Abstract

In view of Nevanlinna theory, we study the properties of systems of two types of complex difference equations with meromorphic solutions. Some results of this paper improve and extend previous theorems given by Gao, and five examples are given to show the extension of solutions of the system of complex difference equations.

MSC:39A50, 30D35.

Keywords

q-shiftdifference equationtranscendental meromorphic function

1 Introduction and main results

In this note, we will investigate the problem of the existence and growth of solutions of complex difference equations. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see [1–3]). Besides, for the meromorphic function f, S(r,f) denotes any quantity satisfying that S(r,f)=o(T(r,f)) for all r outside a possible exceptional set E of finite logarithmic measure limr→∞∫[1,r)∩Edtt<∞, and a meromorphic function a(z) is called a small function with respect to f if T(r,a(z))=S(r,f).

In recent years, difference equations, difference product and q-difference in the complex plane ℂ have been an active topic of study. Considerable attention has been paid to the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory [4–8]. Chiang and Feng [9] and Halburd and Korhonen [10] established a difference analogue of the logarithmic derivative lemma independently. After their work, a number of results on meromorphic solutions of complex difference equations were obtained.

The structure of this paper is as follows. In Section 1, some results on growth of solutions of a complex difference equation are listed, and our theorems are given. In Section 2, we introduce some lemmas. Section 3is devoted to proving Theorem 1.5. Section 4is devoted to proving Theorem 1.6. Finally, Section 5gives some examples to show the accuracy of conclusions of Theorem 1.5.

In 2003, Silvennoinen considered [11] the growth and existence of meromorphic solutions of functional equations of the form f(p(z))=R(z,f(z)), and obtained the following result.

where{J}is a collection of all non-empty subsets of{1,2,…,n}, cj’s are distinct complex constants, andp(z)is a polynomial of degreek≥2. Moreover, we assume that the coefficientsαJ(z)are small functions relative tofand thatn≥k. Then

T(r,f)=O((logr)β+ε),

whereβ=lognlogk.

Recently, there were some paper focusing on the properties of solutions of some systems of complex difference equations and q-shift difference equation (see [12, 13, 15–18]). A question is raised naturally, whether the assertion of Theorem 1.4 remains valid, if the equation (2) is replaced by the following

In this paper, we study the question above and the problem of the existence of meromorphic solutions for a system of complex difference equations (3), where p(z) is a polynomial, and obtain the following results.

Theorem 1.6Under the assumptions of Theorem 1.5, ifp(z)=pkzk+⋯+p1z+p0 (p0,p1,…,pk∈C) of degreek≥2, (f1,f2)is a meromorphic solution of system (3) such thatf1, f2are non-rational meromorphic, and all the coefficients of (3) are small functions relative tof1, f2. Then

Letf(z)be a transcendental meromorphic function, and letp(z)=pkzk+pk−1zk−1+⋯+p1z+p0be a complex polynomial of degreek>0. For given0<δ<|pk|, letλ=|pk|+δ, μ=|pk|−δ, then for givenε>0and for sufficiently larger,

Next, we will prove that k2σ1σ2≤n1n2. Suppose that k2σ1σ2>n1n2, then we can get ς=logn1n2−logσ1σ22logk<1. For sufficiently small ε1>0, we have ς1=ς+ε1<1. This contradicts the condition on the transcendency of f1, f2.

Thus, the proof of Theorem 1.6 is completed.

5 Some examples for Theorem 1.5

The following examples show that the conclusions (4) and (5) in Theorem 1.5 are sharp.

We have μ(f1)+μ(f2)=∞. Thus, it shows that (iii) in Theorem 1.5 is true when σ1=σ2=n1=n2=3, c1=3log2, c2=4log2, q=−1 and η=−3log2.

Declarations

Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This project is supported by the NNSF of China (11301233, 61202313) and the Natural Science foundation of Jiangxi Province in China (No. 2010GQS0119, No. 20122BAB201016 and No. 20132BAB211001). The second author is supported in part by the NNSFC (Nos. 11226089, 11201395, 61271370), Beijing Natural Science Foundation (No. 1132013) and The Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (CIT and TCD20130513).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HYX completed the main part of this article, HYX and ZXX corrected the main theorems. All authors read and approved the final manuscript.

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.